The solution of the global relation for the derivative nonlinear Schrödinger equation on the half-line

We consider initial-boundary value problems for the derivative nonlinear Schrödinger (DNLS) equation on the half-line x>0. In a previous work, we showed that the solution q(x,t) can be expressed in terms of the solution of a Riemann-Hilbert problem with jump condition specified by the initial and boundary values of q(x,t). However, for a well-posed problem, only part of the boundary values can be prescribed; the remaining boundary data cannot be independently specified, but are determined by the so-called global relation. In general, an effective solution of the problem therefore requires solving the global relation. Here, we present the solution of the global relation in terms of the solution of a system of nonlinear integral equations. This also provides a construction of the Dirichlet-to-Neumann map for the DNLS equation on the half-line.     

Robin boundary condition and shock problem for the focusing nonlinear Schrödinger equation

We consider the initial boundary value (IBV) problem for the focusing nonlinear Schrödinger equation in the quarter plane x>0, t >0 in the case of periodic initial data, u(x,0) = α exp(?2iβx) (or asymptotically periodic, u(x, 0) =α exp(?2iβx)→0 as x→∞), and a Robin boundary condition at x = 0: u_x(0, t)+qu(0, t) = 0, q ≠ 0. Our approach is based on the unified transform (the Fokas method) combined with symmetry considerations for the corresponding Riemann-Hilbert (RH) problems. We present the representation of the solution of the IBV problem in terms of the solution of an associated RH problem. This representation also allows us to determine an initial value (IV) problem, of a shock type, a solution of which being restricted to the half-line x > 0 is the solution of the original IBV problem. In the case β < 0, the large-time asymptotics of the solution of the IBV problem is presented in the “rarefaction” sector, demonstrating, in particular, an oscillatory behavior of the boundary values in the case q > 0, contrary to the decay to 0 in the case q < 0.     

On the initial-boundary value problems for soliton equations

We present a novel approach to solving initial-boundary value problems on the segment and the half line for soliton equations. Our method is illustrated by solving a prototypal and widely applied dispersive soliton equation—the celebrated nonlinear Schroedinger equation. It is well known that the basic difficulty associated with boundaries is that some coefficients of the evolution equation of the (x) scattering matrix S(k, t) depend on unknown boundary data. In this paper, we overcome this difficulty by expressing the unknown boundary data in terms of elements of the scattering matrix itself to obtain a nonlinear integrodifferential evolution equation for S(k, t). We also sketch an alternative approach in the semiline case on the basis of a nonlinear equation for S(k, t), which does not contain unknown boundary data; in this way, the “linearizable” boundary value problems correspond to the cases in which S(k, t) can be found by solving a linear Riemann-Hilbert problem.     

Initial-boundary value problem for the two-component Gerdjikov-Ivanov equation on the interval

In this paper, we apply Fokas unified method to study initial-boundary value problems for the two-component Gerdjikov-Ivanov equation formulated on the finite interval with 3×3 Lax pairs. The solution can be expressed in terms of the solution of a 3×3 Riemann-Hilbert problem. The relevant jump matrices are explicitly given in terms of three matrix-value spectral functions s (λ), S (λ) and S_L(λ), which arising from the initial values at t = 0, boundary values at x = 0 and boundary values at x = L, respectively. Moreover, The associated Dirichlet to Neumann map is analyzed via the global relation. The relevant formulae for boundary value problems on the finite interval can reduce to ones on the half-line as the length of the interval tends to infinity.     

The Initial-boundary Value Problem for the Ostrovsky-Vakhnenko Equation on the Half-line

We analyze an initial-boundary value problem for the Ostrovsky-Vakhnenko equation on the half-line. This equation can be viewed as the short wave model for the Degasperis-Procesi (DP) equation. We show that the solution u(x,t) can be recovered from its initial and boundary values via the solution of a vector Riemann-Hilbert problem formulated in the plane of a complex spectral parameter z.     

Integrable Nonlinear Evolution Equations on a Finite Interval

Let q(x,t) satisfy an integrable nonlinear evolution PDE on the interval 0<x<L, and let the order of the highest x-derivative be n. For a problem to be at least linearly well-posed one must prescribe N boundary conditions at x=0 and n−N boundary conditions at x=L, where if n is even, N=n/2, and if n is odd, N is either (n−1)/2 or (n+1)/2, depending on the sign of ∂ⁿ_xq. For example, for the sine-Gordon (sG) equation one must prescribe one boundary condition at each end, while for the modified Korteweg-de Vries (mKdV) equations involving q_t+q_xxx and q_t−q_xxx one must prescribe one and two boundary conditions, respectively, at x=0. We will refer to these two mKdV equations as mKdV-I and mKdV-II, respectively. Here we analyze the Dirichlet problem for the sG equation, as well as typical boundary value problems for the mKdV-I and mKdV-II equations. We first show that the unknown boundary values at each end (for example, q_x(0,t) and q_x(L,t) in the case of the Dirichlet problem for the sG equation) can be expressed in terms of the given initial and boundary conditions through a system of four nonlinear ODEs. We then show that q(x,t) can be expressed in terms of the solution of a 2×2 matrix Riemann-Hilbert problem formulated in the complex k-plane. This problem has explicit (x,t) dependence in the form of an exponential; for example, for the case of the sG this exponential is exp {i(k−1/k)x+i(k+1/k)t}. Furthermore, the relevant jump matrices are explicitly given in terms of the spectral functions {a(k),b(k)}, {A(k),B(k)}, and , which in turn are defined in terms of the initial conditions, of the boundary values of q and of its x-derivatives at x=0, and of the boundary values of q and of its x-derivatives at x=L, respectively. This Riemann-Hilbert problem has a global solution.     

The Modified Korteweg-de Vries Equation on the Half-Line with a Sine-Wave as Dirichlet Datum

Boundary value problems for integrable nonlinear evolution PDEs, like the modified KdV equation, formulated on the half-line can be analyzed by the so-called unified transform method. For the modified KdV equation, this method yields the solution in terms of the solution of a matrix Riemann-Hilbert problem uniquely determined in terms of the initial datum q(x,0), as well as of the boundary values {q(0, t),qx(0, t),qxx(0, t)}. For the Dirichlet problem, it is necessary to characterize the unknown boundary values qx(0, t) and qxx(0, t) in terms of the given data q(x, 0) and q(0, t). It is shown here that in the particular case of a vanishing initial datum and of a sine wave as Dirichlet datum, qx(0, t) and qxx(0, t) can be computed explicitly at least up to third order in a perturbative expansion and that at least up to this order, these functions are asymptotically periodic for large t.     

The derivative nonlinear Schrödinger equation on the half-line

We analyze the derivative nonlinear Schrödinger equation on the half-line using the Fokas method. Assuming that the solution q(x,t) exists, we show that it can be represented in terms of the solution of a matrix Riemann-Hilbert problem formulated in the plane of the complex spectral parameter ζ. The jump matrix has explicit x,t dependence and is given in terms of the spectral functions a(ζ), b(ζ) (obtained from the initial data q₀(x)=q(x,0)) as well as A(ζ), B(ζ) (obtained from the boundary values g₀(t)=q(0,t) and g₁(t)=q_x(0,t)). The spectral functions are not independent, but related by a compatibility condition, the so-called global relation. Given initial and boundary values {q₀(x),g₀(t),g₁(t)} such that there exist spectral functions satisfying the global relation, we show that the function q(x,t) defined by the above Riemann-Hilbert problem exists globally and solves the derivative nonlinear Schrödinger equation with the prescribed initial and boundary values.     

Initial-Boundary Value Problem for the Camassa�CHolm Equation with Linearizable Boundary Condition

We present a Riemann?CHilbert problem formalism for the initial boundary value problem for the Camassa?CHolm equation on the half-line x > 0 with homogeneous Dirichlet boundary condition at x = 0. We show that, similarly to the problem on the whole line, the solution of this problem can be obtained in parametric form via the solution of a Riemann?CHilbert problem determined by the initial data via associated spectral functions. This allows us to apply the non-linear steepest descent method and to describe the large-time asymptotics of the solution.     

Riemann-Hilbert problems for the Ernst equation and fibre bundles

Riemann-Hilbert techniques are used in the theory of completely integrable differential equations to generate solutions that contain a free function which can be used at least in principle to solve initial or boundary-value problems. The solution of a boundary-value problem is thus reduced to the identification of the jump data of the Riemann-Hilbert problem from the boundary data. But even if this can be achieved, it is very difficult to get explicit solutions since the matrix Riemann-Hilbert problem is equivalent to an integral equation. In the case of the Ernst equation (the stationary axisymmetric Einstein equations in vacuum), it was shown in a previous work that the matrix problem is gauge equivalent to a scalar problem on a Riemann surface. If the jump data of the original problem are rational functions, this surface will be compact which makes it possible to give explicit solutions in terms of hyperelliptic theta functions. In the present work, we discuss Riemann-Hilbert problems on Riemann surfaces in the framework of fibre bundles. This makes it possible to treat the compact and the non-compact case in the same setting and to apply general existence theorems.     

Initial value problem of the Whitham equations for the Camassa-Holm equation

We study the Whitham equations for the Camassa-Holm equation. The equations are neither strictly hyperbolic nor genuinely nonlinear. We are interested in the initial value problem of the Whitham equations. When the initial values are given by a step function, the Whitham solution is self-similar. When the initial values are given by a smooth function, the Whitham solution exists within a cusp in the x-t plane. On the boundary of the cusp, the Whitham solution matches the Burgers solution, which exists outside the cusp.     

A method of linearization for Painlevé equations: Painlevé IV, V

A method to linearize the initial value problem of the Painlevé equations IV, V is given. The procedure involves formulating a Riemann-Hilbert boundary value problem on intersecting lines for the inverse monodromy problem. This boundary value problem is reduced to a sequence of standard problems on single lines in a certain range of parameter space. Schlesinger transformations allow one to completely cover the parameter space. Special solutions are constructed from special cases of the Riemann problem as well.     

On solutions of a boundary value problem for a polyharmonic equation in unbounded domains

In the paper, we study uniqueness problems for solutions of a boundary value problem for a polyharmonic equation in the exterior of a compact set and in a half-space under the assumption that the generalized solution of the problem in question admits a finite Dirichlet integral with a weight of the form |x| ^a . In dependence on the values of the parameter a, we prove uniqueness theorems and also present precise formulas to evaluate the dimension of the space of solutions of this problem in the exterior of a compact set and in a half-space.     

Exact solutions for nonlinear fractional anomalous diffusion equations

This work is devoted to investigating exact solutions of generalized nonlinear fractional diffusion equations with external force and absorption. We first investigate the nonlinear anomalous diffusion equations with one-fractional derivative and then multi-fractional ones. In both situations, we obtain the corresponding exact solution, its diffusive behavior, and the sufficient and necessary conditions for solutions satisfying the boundary condition W(±∞,t)=0 and the sharp initial condition W(x,0)=δ(x).     

Two-dimensional wave equation with degeneration on the curvilinear boundary of the domain and asymptotic solutions with localized initial data

In a two-dimensional domain Ω ? R ², we consider the wave equation with variable velocity c(x ₁, x ₂) degenerating on the boundary Γ = ?Ω as the square root of the distance to the boundary, and construct an asymptotic solution of the Cauchy problem with localized initial data. This problem is related to the so-called “run-up problem” in tsunami wave theory. One main idea (also used by the authors in earlier papers in the one-dimensional case and the two-dimensional case with c ²(x ₁, x ₂) = x ₁) is that the (singular) curve Γ is a caustic of special type. We use this idea to introduce a generalization of the Maslov canonical operator covering the problem with degeneration and obtain efficient formulas for the asymptotic solutions.     

Non-Intersecting Squared Bessel Paths at a Hard-Edge Tacnode

The squared Bessel process is a 1-dimensional diffusion process related to the squared norm of a higher dimensional Brownian motion. We study a model of n non-intersecting squared Bessel paths, with all paths starting at the same point a > 0 at time t = 0 and ending at the same point b > 0 at time t = 1. Our interest lies in the critical regime ab = 1/4, for which the paths are tangent to the hard edge at the origin at a critical time ${t^*\in (0,1)}$ . The critical behavior of the paths for n → ∞ is studied in a scaling limit with time t = t ^* + O(n ^?1/3) and temperature T = 1 + O(n ^?2/3). This leads to a critical correlation kernel that is defined via a new Riemann-Hilbert problem of size 4 × 4. The Riemann-Hilbert problem gives rise to a new Lax pair representation for the Hastings-McLeod solution to the inhomogeneous Painlevé II equation q′′(x) = xq(x) + 2q ³(x) ? ν, where ν = α + 1/2 with α > ?1 the parameter of the squared Bessel process. These results extend our recent work with Kuijlaars and Zhang (Comm Pure Appl Math 64:1305–1383, 2011) for the homogeneous case ν = 0.     

On the initial value problem of the second Painlevé Transcendent

The initial value problem associated with the second Painlevé Transcendent is linearized via a matrix, discontinuous, homogeneous Riemann-Hilbert (RH) problem defined on a complicated contour (six rays intersecting at the origin). This problem is mapped through a series of transformations to three different simple Riemann-Hilbert problems, each of which can be solved via a system of two Fredholm integral equations. The connection of these results with the inverse scattering transform in one and two dimensions is also pointed out.     

Method for solving moving boundary value problems for linear evolution equations

We introduce a method of solving initial boundary value problems for linear evolution equations in a time-dependent domain, and we apply it to an equation with dispersion relation omega(k), in the domain l(t)相似文献   

Nonlinear Steepest Descent Asymptotics for Semiclassical Limit of Integrable Systems: Continuation in the Parameter Space

The initial value problem for an integrable system, such as the Nonlinear Schrödinger equation, is solved by subjecting the linear eigenvalue problem arising from its Lax pair to inverse scattering, and, thus, transforming it to a matrix Riemann-Hilbert problem (RHP) in the spectral variable. In the semiclassical limit, the method of nonlinear steepest descent ([4,5]), supplemented by the g-function mechanism ([3]), is applied to this RHP to produce explicit asymptotic solution formulae for the integrable system. These formule are based on a hyperelliptic Riemann surface ${\mathcal {R} = \mathcal {R}(x,t)}The initial value problem for an integrable system, such as the Nonlinear Schr?dinger equation, is solved by subjecting the linear eigenvalue problem arising from its Lax pair to inverse scattering, and, thus, transforming it to a matrix Riemann-Hilbert problem (RHP) in the spectral variable. In the semiclassical limit, the method of nonlinear steepest descent ([4,5]), supplemented by the g-function mechanism ([3]), is applied to this RHP to produce explicit asymptotic solution formulae for the integrable system. These formule are based on a hyperelliptic Riemann surface R = R(x,t){\mathcal {R} = \mathcal {R}(x,t)} in the spectral variable, where the space-time variables (x, t) play the role of external parameters. The curves in the x, t plane, separating regions of different genuses of R(x,t){\mathcal {R}(x,t)}, are called breaking curves or nonlinear caustics. The genus of R(x,t){\mathcal {R}(x,t)} is related to the number of oscillatory phases in the asymptotic solution of the integrable system at the point x, t. The evolution theorem ([10]) guarantees continuous evolution of the asymptotic solution in the space-time away from the breaking curves. In the case of the analytic scattering data f(z; x, t) (in the NLS case, f is a normalized logarithm of the reflection coefficient with time evolution included), the primary role in the breaking mechanism is played by a phase function á h(z;x,t){{\Im\,h(z;x,t)}}, which is closely related to the g function. Namely, a break can be caused ([10]) either through the change of topology of zero level curves of á h(z;x,t){\Im\,h(z;x,t)} (regular break), or through the interaction of zero level curves of á h(z;x,t){{\Im\,h(z;x,t)}} with singularities of f (singular break). Every time a breaking curve in the x, t plane is reached, one has to prove the validity of the nonlinear steepest descent asymptotics in the region across the curve.     

Non-Intersecting Squared Bessel Paths and Multiple Orthogonal Polynomials for Modified Bessel Weights

We study a model of n non-intersecting squared Bessel processes in the confluent case: all paths start at time t = 0 at the same positive value x = a, remain positive, and are conditioned to end at time t = T at x = 0. In the limit n → ∞, after appropriate rescaling, the paths fill out a region in the tx-plane that we describe explicitly. In particular, the paths initially stay away from the hard edge at x = 0, but at a certain critical time t* the smallest paths hit the hard edge and from then on are stuck to it. For t ≠ t* we obtain the usual scaling limits from random matrix theory, namely the sine, Airy, and Bessel kernels. A key fact is that the positions of the paths at any time t constitute a multiple orthogonal polynomial ensemble, corresponding to a system of two modified Bessel-type weights. As a consequence, there is a 3 × 3 matrix valued Riemann-Hilbert problem characterizing this model, that we analyze in the large n limit using the Deift-Zhou steepest descent method. There are some novel ingredients in the Riemann-Hilbert analysis that are of independent interest.